diff --git a/research/gaussian-global-tail/README.md b/research/gaussian-global-tail/README.md new file mode 100644 index 0000000000..f6483fcb36 --- /dev/null +++ b/research/gaussian-global-tail/README.md @@ -0,0 +1,116 @@ +# The written global tail is p^{10} exp(-p^2) + +**Claim type:** a comparison of three tail statements. +**No new κ.** The ideal-gap gauge stays a hypothesis. The selected record +remains `23/10000`. + +## Question + +#4520 asks whether `||P_glob - C_i||_2` is below `p^6 exp(-p^2)`, or only +the written `p^{10} exp(-p^2)`. The named step is to compare the identity +`P - H^{-1} = -Q^J H^{-1}` with the within-cell `p^6` bound, and not to put +`p^{10}` into the gate yet. + +## Lemma + +Under HALO's written hypotheses, the global majorant is +`p^{10} exp(-p^2)`. It is not the within-cell bound `p^6 exp(-p^2)`. + +1. **The identity keeps the inverse factor.** The locality audit records + +```text +P - H^{-1} = -Q^J H^{-1}, +||P - H^{-1}||_2 ≤ (1 - p^{-40})^J p^{10} ≤ p^{10} exp(-p^2). +``` + +For an integer `p`, `J = ceil(p^{42}) = p^{42}`, so `J p^{-40} = p^2`. +That power fills the `exp(-p^2)` slot. It does not cancel any power of the +inverse factor. HALO states the same majorant, +`||P_i - H_i^{-1}||_2 ≤ p^{10} exp(-p^2)`. + +2. **The written majorant is larger than `p^6 exp(-p^2)` by `p^4`.** +Since `10 - 6 = 4` and `p^4 ≥ 16 > 1` for every integer `p ≥ 2`, + +```text +p^{10} exp(-p^2) = p^4 · p^6 exp(-p^2) > p^6 exp(-p^2). +``` + +The truncation exponent `2` and the leftover exponent `4` are different. +The series length does not turn the written majorant into the smaller one. + +3. **The within-cell `p^6` bound is a different polynomial.** #4452 proves +`||P - A^{-1}||_2 < p^6 exp(-p^2)` for +`P = p^{-11} sum_{j < p^{24}} Q^j A^*` inside one cell, under the ideal-gap +gauge. That package does not move the global reference. + +4. **The conditional `p^6` bound is not the written hypothesis.** #4406 +proves `||P - H^{-1}||_2 < p^6 exp(-p^2)` for +`P = p^{-12} sum_{j < p^{26}} Q^j H^*`, and only if +`||H^{-1}||_2 ≤ 16 p^2`. It does not prove that ceiling for the rounded +reference. At `p = 2`, `16 p^2 = p^6`. For `p ≥ 3`, `16 p^2 < p^6`. +Equality at `p = 2` means the ceiling `16 p^2` is not by itself a strict +`p^6` bound. + +This package does not put `p^{10}` into the #4520 gate. + +## Proof + +The audit derives the displayed identity from +`p^{-20} (I - Q)^{-1} H^* = H^{-1}` and commutation of `Q` with `H^* H`. +The factor `p^{10}` is the hypothesis `||H^{-1}||_2 ≤ p^{10}`. Multiplying +by `(1 - p^{-40})^J` and using `J p^{-40} = p^2` produces the written +majorant. Replacing `p^{10}` by `p^6` in that product would require +`||H^{-1}||_2 ≤ p^6`, which is not HALO's hypothesis. + +The integer comparison is `2^{10} = 2^6 · 2^4` and `2^4 > 1`. The same +gap is `p^4` at every larger integer `p`. The conditional ceiling meets +`p^6` at `p = 2` and is smaller at `p = 3`, where `16 · 9 < 3^6`. + +## Not covered + +- A proof that `||H^{-1}||_2 ≤ p^{10}` or `||H^{-1}||_2 ≤ 16 p^2` for the + rounded reference. Both remain hypotheses of the packages that use them. +- The ideal-gap gauge behind the within-cell bound. +- Absorbing `p^{10}` in the restricted gate. +- Editing the #4100 notes. They still say `p^{10} exp(-p^2)`. +- Any new κ. + +## Reproduce + +```sh +python3.12 -B scripts/research_package.py --package research/gaussian-global-tail --check-only +python3.12 -B scripts/research_package.py --package research/gaussian-global-tail +``` + +Base `f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc`, priority revision 7, selection +refresh κ = `23/10000`. Upstream was fetched at 2026-10-22T03:23:31Z and was +that commit. + +Overlap snapshot: open PRs fetched 2026-10-22T03:25:01Z (308 records) and +discussions fetched 2026-10-22T03:25:44Z (3401 records). A search for +`global tail`, `does not move the global reference` and `p^10 exp(-p^2)` +returned #4520, which asks this question, the locality audit, and #4452. +The other hits use "global tail" for unrelated transforms. No result +replaces HALO's `p^{10}` majorant by the within-cell `p^6` bound. A search +miss is not a novelty certificate. The audit is the identity. #4452 is the +within-cell bound. HALO is the written tail. + +## Open next questions + +1. **Does the restricted gate absorb one section error `p^6 exp(-p^2)` and + one global tail `p^{10} exp(-p^2)`?** The two errors are not the same + `ε`. Their sum is `p^6 (1 + p^4) exp(-p^2)`, not `2 p^6 exp(-p^2)`. The + first step is to place the factor `1 + p^4` in the exponent and clear + the eighths. HALO already assumes `||H^{-1}||_2 ≤ p^{10}`; this question + does not ask for a proof of that assumption. + +## Credit + +- The identity and the factor `p^{10}` are the locality audit's, applied to + HALO's polynomial. +- The within-cell `p^6` bound, and the refusal to move the global reference, + are #4452's. +- The conditional `p^6` bound under `||H^{-1}||_2 ≤ 16 p^2` is #4406's. +- The question is #4520's. + +Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0. diff --git a/research/gaussian-global-tail/SOURCES.json b/research/gaussian-global-tail/SOURCES.json new file mode 100644 index 0000000000..69dadd2851 --- /dev/null +++ b/research/gaussian-global-tail/SOURCES.json @@ -0,0 +1,42 @@ +{ + "schema_version": 1, + "base_commit": "f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40", + "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65", + "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96" + }, + { + "path": "research/gaussian-delta-exponents/README.md", + "usage": "input", + "git_blob": "6e5f1cc5b77a2ddefaf6f3aba658535e506b3346", + "sha256": "e7c954e592a34df777a6e888c72b3fdf6968abf8cf7b3419fa447ee716142989" + }, + { + "path": "research/gaussian-halo-exponents/README.md", + "usage": "input", + "git_blob": "4f70baafa951df301b4d8e92fe0aff1fd58b2f6e", + "sha256": "5d6c75806e51605c56c8dd29a58b5e5f9b6e6e475d4a1598d55ca341db171f5d" + }, + { + "path": "research/halo-locality-witness-audit/README.md", + "usage": "input", + "git_blob": "03b244baf61b427edbccd29698a05583b5405dd6", + "sha256": "fc87657cb3e89a74b9462483cc15713f0122256a623fefa65c887e2118f81276" + }, + { + "path": "research/unconditional-batched-crt/HALO.md", + "usage": "input", + "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b", + "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc" + } + ] +} diff --git a/research/gaussian-global-tail/TASK.json b/research/gaussian-global-tail/TASK.json new file mode 100644 index 0000000000..e5925f2fe0 --- /dev/null +++ b/research/gaussian-global-tail/TASK.json @@ -0,0 +1,74 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 7, + "base_commit": "f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc", + "original_card": { + "id": "uniform-transfer-audit", + "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?", + "starting_points": [ + "docs/research/research-family-status.md", + "certificates/selected-result.json", + "research/unconditional-batched-crt/README.md" + ], + "methods": [ + "Prove one missing implication with explicit quantified hypotheses.", + "Find an exact counterexample at the claimed interface boundary." + ], + "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.", + "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately." + }, + "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.", + "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.", + "allowed_variations": [ + "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller." + ], + "scope_change_triggers": [ + "Replacing a quantified obligation with another finite smoke test.", + "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses." + ], + "overlap_terms": [ + "uniform transfer", + "all-size", + "precision", + "4100", + "unconditional-batched-crt" + ], + "existing_checks": [ + { + "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.", + "evidence": "research/unconditional-batched-crt/README.md", + "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.", + "status": "author-checked", + "source_commit": "337d04939571a637aeab54a0f58f54fde811663a", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + } + ] + }, + "scope_status": "refined", + "actual_scope": "Compare HALO's global identity P - H^{-1} = -Q^J H^{-1} with the within-cell p^6 bound, and record that the written global majorant stays p^{10} exp(-p^2).", + "scope_change_reason": "The card asks for one caller-bound precision implication. This package only identifies which tail the written hypotheses prove. It does not put p^{10} into the gate, does not prove the rounded inverse bound, and does not claim a new kappa.", + "bottleneck_link": "The selected frontier is unchanged. This is the global-tail question left open by the restricted monomial.", + "budget_minutes": 60, + "stop_rule": "Stop once the written global majorant is separated from the within-cell p^6 bound. Do not absorb p^{10} in the gate, do not prove ||H^{-1}||_2 <= 16 p^2, do not edit the #4100 notes, and do not claim a new kappa.", + "outcome": "completed", + "evidence": "python3.12 -B scripts/research_package.py --package research/gaussian-global-tail", + "author": "James Reed", + "ai_assistance": "Substantial help from Cursor Grok 4.7 on the tail comparison and the package text.", + "open_questions": [ + { + "question": "Does the restricted gate absorb one section error p^6 exp(-p^2) and one global tail p^{10} exp(-p^2)?", + "why_it_matters": "The two errors are not the same epsilon. Their sum is p^6(1 + p^4) exp(-p^2), not 2 p^6 exp(-p^2).", + "first_step": "Place the factor 1 + p^4 in the exponent of the restricted monomial and clear the eighths. Do not prove ||H^{-1}||_2 <= p^{10}; HALO already assumes it.", + "blocked_by": "The section p^6 bound still needs the ideal-gap gauge, and the conditional p^6 ceiling needs ||H^{-1}||_2 <= 16 p^2 for the rounded reference." + } + ] +} diff --git a/research/gaussian-global-tail/VERIFY.json b/research/gaussian-global-tail/VERIFY.json new file mode 100644 index 0000000000..498cc75f18 --- /dev/null +++ b/research/gaussian-global-tail/VERIFY.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "certificates/selected-result.json", + "research/gaussian-delta-exponents/README.md", + "research/gaussian-halo-exponents/README.md", + "research/halo-locality-witness-audit/README.md", + "research/unconditional-batched-crt/HALO.md" + ], + "commands": [ + { + "argv": [ + "{python}", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "HALO's truncation uses p^{42} against p^{-40}, which supplies the exp(-p^2) slot and leaves the inverse factor p^{10}. That factor exceeds p^6 by p^4. The within-cell p^6 bound does not move the global reference, and the conditional p^6 bound needs an unproved ceiling of 16 p^2." + } + ] +} diff --git a/research/gaussian-global-tail/verify.py b/research/gaussian-global-tail/verify.py new file mode 100644 index 0000000000..fcbb5036f6 --- /dev/null +++ b/research/gaussian-global-tail/verify.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python3 +"""Separate HALO's written global tail from the within-cell p^6 bound. + +Every comparison is an integer identity. The ideal-gap gauge stays a hypothesis. +""" + +from __future__ import annotations + +import hashlib +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[2] +PACKAGE = ROOT / "research" / "gaussian-global-tail" + + +def digest(path: Path) -> str: + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def pins_and_phrases() -> None: + sources = json.loads((PACKAGE / "SOURCES.json").read_text(encoding="utf-8")) + for row in sources["files"]: + if row["usage"] == "input": + assert digest(ROOT / row["path"]) == row["sha256"], row["path"] + halo = (ROOT / "research/unconditional-batched-crt/HALO.md").read_text(encoding="utf-8") + audit = (ROOT / "research/halo-locality-witness-audit/README.md").read_text(encoding="utf-8") + cell = (ROOT / "research/gaussian-delta-exponents/README.md").read_text(encoding="utf-8") + conditional = (ROOT / "research/gaussian-halo-exponents/README.md").read_text(encoding="utf-8") + selected = json.loads( + (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8") + ) + assert "||P_i-H_i^-1||_2 <= p^10 exp(-p^2)." in halo + assert "P - H^-1 = -Q^J H^-1." in audit + assert "(1-p^-40)^J p^10 <= exp(-J p^-40) p^10 <= p^10 exp(-p^2)." in audit + assert "||P - A^{-1}||_2 < p^6 exp(-p^2)" in cell + assert "does not move the global reference." in cell + assert "||P - H^-1||_2 < p^6 exp(-p^2)," in conditional + assert "It does not prove `||H^-1||_2 <= 16 p^2` for the rounded reference." in conditional + assert selected["kappa"] == "23/10000" + + +def tail_gap() -> None: + # J = p^{42} against p^{-40} fills exactly the exp(-p^2) slot. + assert 42 - 40 == 2 + # The written inverse factor is larger than p^6 by p^4. + assert 10 - 6 == 4 + assert 2 != 4 + assert 2**4 == 16 + assert 16 > 1 + assert 2**10 == 2**6 * 2**4 + # At p = 2 the conditional ceiling 16 p^2 equals p^6. It is smaller later. + assert 16 * 2**2 == 2**6 + assert 16 * 3**2 < 3**6 + + +def main() -> None: + pins_and_phrases() + tail_gap() + print("PASS_GAUSSIAN_GLOBAL_TAIL") + + +if __name__ == "__main__": + main()